Expected Value and Variance

X is continuous if it can take any value from one or more intervals of teal numbers. We cannot use a p.m.f. to describe the probability distribution of X, because its possible values are uncountably infinite. We use a new notion called the probability density function (p.d.f) such that areas under the/(x) curve represent probabilities. The p.d.f. of a continuous r.v. X is the function that satisfies 

Example 2.9 Probability Calculation from Exponential Distribution. The simplest distribution used to model the times to failure (lifetime) of items or survival times of patients is the exponential distribution. The p.d.f. of the exponential distribution is given by 

completely describes the probabilistic behavior of a r.v. However, certain numerical measures computed fi om the distribution provide useful summaries. 

This is a sum of possible values, xj, X2. taken by the r.v. X weighted by their probabilities. 

Example 2.10 Expectation of Discrete Random Variable. Suppose two dice are tossed (Example 2.8). Let Xi be the sum of two dice and X2 and X3 be the values of the first and second tossings, respectively. What are the expectations of Xi, X2 + X3  

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Formulas are available for the computation of the expected value and variance of the ZIP random variable (18). The expected value for a ZIP-distributed random variable Y, E(Y), is... 

The Pegg-Barnett Hermitian phase formalism allows for direct calculations of quantum phase properties of optical fields. As the Hermitian phase operator is defined, one can calculate the expectation value and variance of this operator for a given state /). Moreover, the Pegg-Barnett phase formalism allows for the introduction of the continuous phase probability distribution, which is a representation of the quantum state of the field and describes the phase properties of the field in a very spectacular fashion. For so-called physical states, that is, states of finite energy, the Pegg-Barnett formalism simplifies considerably. In the limit as a —> oo one can introduce the continuous phase distribution... 

If g(0), be it univariate or multivariate, is a nonlinear function then an approach repeatedly seen throughout this book will be used—the function will first be linearized using a first-order Taylor series and then the expected value and variance will be found using Eqs. (3.55) and (3.56), respectively. This is the so-called delta method. If g(0) is a univariate, nonlinear function then to a first-order Taylor series approximation about 0 would be... 

Knowing the pdf of a distribution is useful because then the expected value and variance of the distribution may be derived. The expected value of a pdf is its mean and can be thought of as the center of mass of the density. If X is a continuous random with domain — oo, oo variable then its expected value is... 

Many families of probability distributions depend on only a few parameters. Collectively, these parameters will be referred to as 0, the population parameters, because they describe the distribution. For example, the exponential distribution depends only on the parameter X as the population mean is equal to 1/X and the variance is equal to 1/X2 [see Eqs. (A.50) and (A.51)]. Most probability distributions are summarized by the first two moments of the distribution. The first moment is a measure of central tendency, the mean, which is also called the expected value or location parameter. The second moment is a measure of the dispersion around the mean and is called the variance of the distribution or scale parameter. Given a random variable Y, the expected value and variance of Y will be written as E(Y) and Var(Y), respectively. Unless there is some a priori knowledge of these values, they must be estimated from observed data. 

This is a discrete distribution, over the non-negative integers. Its expected value and variance are equal ... 

In addition to expected value and variance the distribution percentiles are used to characteri2e a distribution. The percentiles are values below which a certain fraction of the distribution lies. In use are the 5th, 50th (median) and 95th percentiles. Using Eq. (C.17) we obtain for continuous random variables... 

General theorems exist in the mathematics of statistics to predict the means and variances for probability density functions of arbitrary form. These theorems may be applied where the probability density function is not known or is inconvenient to calculate. It can be shown [42] for the difference expressed by Eq. (4.87) that the expectation values and variances are related by... 

If Fi, F2,..., Fj. are a sequence of independent random variables having the same N fi, a distribution with exactly known expected value and variance, then the sum of squares of the standardized variables will have a x k) distribution. 

The first and second moments-expected value and variance-of the adjusted prediction, y, are calculated by Eqs. 11 and 12 which fully define a normal distribution representing the xmcertainty inherent in the prediction of a system response as a result of model-form xmcertainty. 

This multiplicative adjustment factor, E, is assumed to be a log normally distributed factor representing the xmcertainty present in the selection of the best model as being most accurate at predicting the true physical response. In assuming a log normally distributed form for this factor, the first and second moments-expected value and variance-of the adjustment factor are calculated as shown in Eqs. 14 and 15. 

Equation 18 is similar to Eq. 8 from the additive adjustment factor approach with the slight difference in the utilization of the expected value of the best model, E[y ], rather than the deterministic best model prediction. Calculating the first and second moments-expected value and variance-of the probabilistic adjustment factor is also different for this new approach, as the approach had to be rederived to handle the stochastic model set. The calculation of these... 

After calculating the first and second moments of the probabilistic adjustment factor, the expected value and variance of the adjusted model can then be calculated using Eqs. 21 and 22. 

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